3rd Semester VECTOR ANALYSIS PROGRAM STUDI PENDIDIKAN MATEMATIKA FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN UNIVERSITAS MUHAMMADIYAH SURAKARTA

Credits : 2 credits Lecturer : Sri Rejeki Phone Numb : 085725313171 E mail : sri.rejeki@ums.ac.id Website : http//www.srirejeki171.wordpress.com

OVERVIEW OF THE COURSE CONTENTS COURSE OBJECTIVES Students are expected to be able to apply the concepts of vector and vector calculus to solve problems which they found in daily life. OVERVIEW OF THE COURSE CONTENTS This course is a two credits course which is given to students in the second semester. The main topics discussed in this course are: the consepts of vector and scalar, the basic operations of vectors, vector function, the integration and derivative of vector fuctions, vector field and gradient of vector field, curve integral, surface integral, Curl and Divergence, Green Theorem, Stokes Theorem, aplikasi teorema Stokes, and curvelinear coordinates.

LEARNING OBJECTIVES Applying the concepts of scalar and vector Applying the derivative of vector fuction to solve problems Determining vector fields and gradients of vector field Applying the concepts of curve integral to solve problems Applying the concepts of surface integral to solve problems Applying the concepts of Curl and Divergence to solve problems Applying the concepts of Green Theorem to solve problems Applying the concepts of Stokes Theorem to solve problems Determining curvelinear coordinates

SOURCES Janich, K. 2000. Vektor Analysis. Springer: New York. James, S. 2012. Multivariable Calculus 7th Edition. Sukirman. 2007. Analisis Vektor. Universitas Terbuka: Jakarta. Spiegel, M. R. 1991. Analisis Vektor. Seri Buku Schaum. Erlangga: Jakarta.

ASSESMENT Grade : score >= 77 A 70 =< score < 77 AB 63 =< score < 70 B 56 =< score < 63 BC 50 =< score < 56 C 35 =< score < 50 D score < 35 E

LEARNING PLAN WEEK LEARNING ACTIVITIES MATERI KULIAH DAN BENTUK KEGIATAN 1 2 3 4 5 6 7 Introduction Lecturing&Discussion Didactical Contracts The concepts of vector and scalar Dot products and cross products Line Equation and Plane Equation Cylinder and quadratic surfaces Vector Function and space curve Arc Length and Curvature Derivative and integral of vector functions MID TEST

LEARNING PLAN 8 9 10 11 12 13 14 Lecturing and discussion WEEK LEARNING ACTIVITIES TOPICS 8 9 10 11 12 13 14 Lecturing and discussion Vector fields and gradients Curva Integral Surface integral Green Theorem Curl and divergence Stokes Theorem Curvelinear Coordinates. FINAL TEST

DIDACTICAL CONTRACTS Kegiatan pemebelajaran dimulai pada jam yang telah disepakati, toleransi keterlambatan tidak terbatas. Selama proses pembelajaran berlangsung HP dimatikan/disilent. Pengumpulan tugas ditetapkan sesuai jadwal dan dilakukan sebelum pembelajaran dimulai. Bagi yang mengumpulkan tugas pada hari terakhir pengumpulan tugas, nilai maksimal C. Aturan jumlah minimal presensi dalam perkuliahan tetap diberlakukan (75%), termasuk aturan cara berpakaian atau bersepatu. Bagi mahasiswa yang terbukti melakukan kecurangan pada saat UTS atau UAS, pekerjaan UTS atau UAS tersebut tidak akan dikoreksi dan otomatis akan mendapatkan nilai 0.

VECTOR AND SCALAR Definition Vector a quantity has both a magnitude and direction in space. E.g: force, velocity, acceleration Scalar a quantity whose value may be represented by a single (positive or negative) real number. E.g: temperature, mass, density, time Vector algebra is the operations of addition, subtraction, and multiplication in algebra of vectors.

Vector in two dimensions: a = a1, a2 Notation : Vector in two dimensions: a = a1, a2 Vector in three dimensions: a = a1, a2, a3 a1, a2, and a3 are called the components of a. The representation of a = a1, a2 is a straight line of arbitrary A(x, y) to B(x + a1, y + a2). The special representation of a is a straight line from O to P(a1, a2). In this case a is called positional vector of P(a1, a2). y For example: Determine a vector which is represented by a straight line from A(2, -5, 0) to B(-3, 1, 1). B(x+a1, y+ a2) P(a1, a2) A(x, y) x O

The length of vector a = a1, a2 is The length of vector a = a1, a2, a3 is y Addition of Vectors If a = a1, a2 and b = b1, b2, then a + b can be defined as a + b b a For vectors in three dimensions can be determined with the same way. O x

Multiplication of Vectors with Scalar If c scalar and a = a1, a2, then vector ca can be defined as For vectors in three dimensions can be determined with the same way. For example: If a = 4, 0,3 and b = -2, 2, 5, determine vectors a + b, 3b, 2a+ 5b, and . .

The properties of vectors If a, b, and c are vectors in a certain dimension, and k and l are scalars, thus a + b = b + a 5. k(a + b) = ka + kb a + (b + c) = (a + b) + c 6. (k + l)a = ka + la a + 0 = a 7. (kl)a = k(la) a + (-a) = 0 8. 1a = a z Standard basis vector i = 1, 0, 0 j = 0,1, 0 k= 0, 0, 1 k j i y x

If a = a1, a2, a3, then can be written as a = a1, a2, a3 = a1, 0, 0 + 0, a2, 0 + 0, 0, a3 = a1 1, 0, 0 + a2 0, 1, 0 + a3 0, 0, 1  a = a1i + a2 j + a3 k For example: If a = i + 2j – 3k and b = 4j + 5k, determine 2a + 5b in i, j, and k form.

Unit vector is a vector of length 1. For example, i, j and k Unit vector is a vector of length 1. For example, i, j and k. If a is a nonzero vector, then the unit vector of a is For example: Determine the unit vector of a vector 2i + j – 2k.

HOMEWORK Determine vectors and scalars from these following quantities: weight, specific heat, density, volume, speed, calories, momentum, energy, distance. A car moving towards the north as far as 3 miles, then 5 miles to the northeast. Describe this movement graphically and determine the resultant displacement vectors graphically and analytically. Show that the addition of vectors is commutative. Given a = 3, -2, 1, b = 2, -4, -3, c = -1, 2, 2 determine the length of a, a+b+c, dan 2a-3b-5c . Given a = 2, -1, 1, b = 1, 3, -2, c = -2, 1, -3, and d = 3, 2, 5 determine scalars k, l, m so that d=ka+lb+mc